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MATH 101: BEGINNING ALGEBRA COURSE KIT August 2011 Learning Outcomes: The student satisfactorily completing Math 101 will: 1. 2. 3. 4. 5. 6. Perform operations with real numbers and algebraic expressions. Solve linear equations, inequalities, and formulas for specified variables. Graph linear equations and determine the equation of a line. Simplify, evaluate, and perform operations with polynomials. Factor polynomials and solve quadratic equations by factoring. Solve application problems involving the procedures and techniques detailed above. Original Course Kit Authors/Editors: Habib Aghdami, Mark Gollwitzer, Debbie Moran, Hala Nestberg, William Parker, Judy Walden, and Bonnie Mullinix The development of this Course Kit was facilitated through the Unlock Your Future Initiative and funded under Title III grant [insert grant #] Math 101: Beginning Algebra Course Kit pg 2 Introduction to the Course Kit This Course Kit has been developed for use by instructors teaching Beginning Algebra at Greenville Technical College with the aim of helping students attain the learning outcomes identified above. It contains all instructional materials needed to conduct active, applied and practice-oriented lessons including: lesson plans (for all primary instructional sessions and sample review lesson plans) activities & task sheets and supplemental materials (links to videos, presentations, websites) course syllabus course outline(s) (with alternate structures) While originally developed independent of specific learning resources (texts, software…), this Course Kit has been adapted to reference the materials selected through the course redesign and pilot for use beginning Fall 2011: Wright (2011). Developmental Mathematics. Hawkes Learning Systems; Hawkes Learning Systems Software and the Multiview TI30-XS Multiview calculator This Course Kit represents the collaborative work of full-time and part-time Math instructors working together to review effective teaching/learning practices in mathematics and develop, pilot and document our best practices. The core team for this working on both Math 101 and Math 102 Curricular revision from Fall 2009 – Fall 2011 included: William Parker (team lead), Habib Aghdami, Mark Gollwitzer, Judy Walden (implementation lead), Hala Nestberg and Debbie Moran. In addition, during the first year this team benefited from contributions from Jean Steele, Toi Graham (lead 09), Mary Ann Billig, and Lucius Pinkney. This is the product of their dedicated and committed work. Contributing to the Course Kit – The instructional power of this course kit comes from the fact that it is built on the wisdom of practice from many Greenville Tech math instructors combining their decades of experience on what works with GTC students. This Kit is not finished. We need your contributions. From these we will build a kit with multiple alternative and equivalent activities that can be interchanged and adapted to meet the distinct needs of various classes of students. If you have a favorite activity that you believe would fit within this curriculum, please propose it by ______________________________ Math 101: Beginning Algebra Course Kit pg 3 Table of Contents Math 101 Course Outline with Lesson Plan/Activity Links General Daily Lesson Plan with Activity/Teaching Ideas Unit 1: Solving Linear Equations Unit 1 Overview - Teaching Insights & Strategies 1a. Course Introduction & Basic Math Review 1b. Simplifying Algebraic Expressions 1c. Solving Linear Equations in One Variable 1d. Solving Absolute Value Equations 1e. Applications of Linear Equations Unit 2: Linear Inequalities; Graphing Linear Equations in Two Variables Unit 2 Overview - Teaching Insights & Strategies 2a. Solving Linear Inequalities 2b. Cartesian Coordinate System 2c. Graphing Linear Equations in Two Variables 2d. Graphing: Slope-Intercept Form 2e. Graphing: Point-Slope Form 2f. Applications of Two Variable Equations Unit 3: Systems of Linear Equations; Exponents Unit 3 Overview - Teaching Insights & Strategies 3a. Systems of Linear Equations: Solve by Graphing & Substitution 3b. Systems of Linear Equations: Solve by Addition 3c. Applications of Linear Systems 3d. Exponents Math 101: Beginning Algebra Course Kit Unit 4: Polynomials; Factoring Unit 4 Overview - Teaching Insights & Strategies 4a. Introduction to Polynomials & Operations 4b. Operations on Polynomials – Multiplying 4c. Operations on Polynomials - Division 4d. Greatest Common Factor, Factor by Grouping 4e. Factoring Trinomials Review Sessions Team-Based Jeopardy Game Review Session for Final (or Test) Review Session for Test – Option 1 (Standard – Online Test) Review Session for Test – Option 2 (Standard – In-Class Test) Appendices Appendix A: Syllabus - Math 101 Beginning Algebra Appendix B: Course Outlines Appendix C: Activities pg 4 Math 101: Beginning Algebra Course Kit Course Outline pg 5 Math 101 Beginning Algebra Lesson Plans with Activities across 30 sessions Session/ Meeting Lesson Plan 1 1a Introduction Basic Math Review 2 1b Unit 1 – Solving Linear Equations Simplifying and Evaluating Algebraic Expressions Translating English Phrases and Algebraic Expressions 3 1c 4 1c cont. 1d 5 1c 6 7 8 1d Review 9 2a 10 11 12 13 14 15 16 2b 2c 2d 2e 2f Review 17 3a 18 3b 19 3c 20 3d 21 22 Review 23 4a 24 4b 25 26 4c 4d 4d 4e 27 28 29 30 Review Focus Solving Linear Equations in One Variable Solving Linear Equations in One Variable Solving Absolute Value Equations Applications of Linear Equations: Number Problems, Consecutive Integers Working with Formulas (primarily solving formulas for different variables) Applications of Linear Equations: Distance-Rate-Time, Interest, Average Review for test on Unit 1 Unit 1 Test Unit 2 – Linear Inequalities; Graphing Linear Equations in Two Variables Solving Linear Inequalities in One variable Solving Absolute Value Inequalities (simplest cases) The Cartesian Coordinate System Graphing Linear Equations in Two Variables Graphing: Slope-Intercept Form y = mx + b Graphing: Point-Slope Form y – y1 = m(x – x1) Applications of Two Variable Equations Review for test on Unit 2 Unit 2 Test Unit 3 – Systems of Linear Equations; Exponents Systems of Linear Equations: Solve by Graphing Systems of Linear Equations: Solve by Substitution Systems of Linear Equations: Solve by Addition Applications of Linear Systems: D=RT, Numbers, Amounts, Costs, Interest and Mixture Exponents & Scientific Notation Review for test on Unit 3 Unit 3 Test Unit 4 – Polynomials; Factoring Introduction to Polynomials Adding and Subtracting with Polynomials Operations on Polynomials - Multiplying Special Products of Binomials Operations on Polynomials - Division Greatest Common Factor, Factor by Grouping continue - Greatest Common Factor, Factor by Grouping Factoring Trinomials Review for test on Unit 4 Unit 4 Test Review for final exam Final Exam Text Section Assignments / Notes 7.1 – 7.6 Review 7.7 Note: Continue review of 7.1 – 7.6 concepts while covering 7.7 7.8 8.1 : x + b = c and ax = c 8.2: ax + b = c 8.3: ax + b = cx + d Appendix A.4: 8.4 8.5 8.6 Unit 1 Test 8.7 Appendix A.4: 9.1 9.2 9.3 9.4 Supplemental material Unit 2 Test 10.1 10.2 10.3 10.4 10.5 11.1 11.2 Unit 3 Test 11.3 11.4 11.5 11.6 11.7 12.1 12.1 continued 12.2: x2 + bx + c Unit 4 Test Math 101: Beginning Algebra Course Kit pg 6 Developmental Studies Lesson Plan General Daily Lesson Plan Initial Lesson by: Judy Walden (& Bonnie Mullinix & Team) Course/Unit Focus: Math 101 Lesson: All Primary Course Outcome(s): changes with specific lesson Learning Objectives: By the end of the session, learners will: 1. 3. Identify the topic and its relationship to previous and upcoming topics 2. Note: Additional Objectives change for each lesson Practice applying the math skills associated with the lesson topic. Materials: Smart board / white board (Computer lab, where needed and other materials as required) Duration: Length of a class period (example: 75 min) Time 5 min Description/Activity Prior to class, write on the board what section(s) will be covered today and what will be covered in the next class period. Suggested: Post an entry activity or problem that students can do as they enter the class. 5 min Introduce the focus and topic of the lesson. Ask students what was done last time and describe where this lesson falls in the course (relationship to previous topics &/or upcoming topics). Give (/ask for) an example of why this topic is important (real-life applications). Make any announcements that need to be made about upcoming quizzes, team/group responsibilities, etc. [Note: When the announcements are important, remember to update Blackboard and/or email the students as well.] 5-10 min Review of problems from the previous class. Use varied structure/activities to involve and assess students (see below for ideas & vary what you do from class to class, add your own) 50-55 min 5 min Incorporate a combination of techniques and interactive activities to involve students in actively developing math skills, solving problems, working in groups, experiencing concepts, using a variety of learning approaches and styles. Intersperse with brief lecturettes or discussions to highlight key points and/or correct misconceptions. Sum up the important points of the lesson. Restate what will be covered in the next lesson and what assignments students should be working on between classes. Announce any upcoming tests, quizzes or projects. Math 101: Beginning Algebra Course Kit pg 7 Guide & Ideas for Effective Math Lessons: Preparation & Board Use At the beginning of each chapter/section, write the chapter number and name and/or section number and name on the board. Write each new term and definition on the board. If the instructions for a problem are not obvious from the problem itself, then write the instructions on the board as well. Review of Last Lesson Use varied structures/activities to involve and assess students’ knowledge and abilities (see below for ideas & vary what you do from class to class, add your own) List review problems on board prior to class, or Give a quiz (individually, in groups; graded or ungraded) Give a quiz online (for homework/in class (lab); identify and work with the problems that gave the most trouble After the students have had time to work the problems, review responses by: Collecting or peer grading quiz & discuss, or Working problems, showing answers on the board, or Having students work problems on the board, or Have students share/compare their results with classmates (such as in a group), “Think/pair/share” – have them think or work the problem. After they have had time, have them pair up and share their results with their partner, explaining their thinking & problem solving strategies (instructor rotates among pairs listening/helping where needed) Class Activities Use a variety of activities in your class to strategically involve students in actively developing math skills, solving problems, working in groups, experiencing concepts, using a variety of learning approaches and styles. Use a combination of teaching and learning techniques and select activities that are well-matched to the topic and focus of the lesson. Design your lessons to ensure that students become involved in learning rather than simply taking notes on a lecture. Intersperse brief lecturettes or discussions to highlight key points and/or correct misconceptions. Selecting Activities There are many activities that can be selected and techniques that can be used. Select activities that relate to the lesson topic/focus: Small Groups - Have students work a similar problem at their desk. Encourage them to work with a partner Teams - Assign working teams (of 3-4 students) to work together. Build teams based on varying levels of expertise (using early diagnostics or other early quiz or placement information). Use these teams regularly and/or shift teams part way through the semester Jigsaw – Put students into groups where they first work a problem (master it) and then in another group (with all problem types represented) where they share their problem and learn how to solve the others. Math 101: Beginning Algebra Course Kit pg 8 Peer teaching – have students (in pairs or groups) responsible for preparing for and introducing new problems to the class (rotating responsibility). Allow students to select which topic they will be responsible for. Provide some class time and/or support for preparing to present. Projects – Groups/Teams are given a project to complete that demonstrates their knowledge of key skills and concepts Online Software – Individuals work on self-paced activities using online learning software. Many More – insert your ideas for activities here (and share with others!) After the students have completed an activity, make sure to “process” and discuss it to ensure that: Answers and solutions to problems are clear Strategies for working through and solving challenging problems are understood by everyone Questions get answered. Use “Lecturettes” Effectively A lecturette is a short lecture of no more than 15-20 minutes that are targeted to a specific topic or concept. They can be as short as 2-5 minutes and remain particularly effective in guiding students from one step to the next. Lecturettes can be used to introduce a topic, clarify a common/shared misconception (while working on problems), or to consolidate, highlight and underscore concepts or key points following an activity. Whether they come before, during or after activities, use them strategically and pointedly to support the other activities. As you deliver a lecturette use visuals, the board, text and examples to help students gather complete and relevant notes that will guide them on their next step. (Note: Students have different learning styles and not all students will remember things that are communicated only verbally). Provide a clear and organized presentation that helps students see the connection between ideas shared and the lesson topics/focus. Review for Test Include multiple ways for students to review for tests. Just as above, activities including lecturettes Use several and/or change what you do: Use Review Games (e.g. Jeopardy game) Have students work sample problems (individually and/or in groups), asking questions of each other or the instructor o Write a list of problem numbers from the chapter review. o Invite students to write & submit test problems (and answers) Provide incentives: o Students may leave early if they finish all problems (individually or in groups) and show them to the instructor to confirm accuracy. o Agree to use selections from the best problems submitted by students in the test o Provide participation points Conclusion Use your last few minutes of class to sum up the important points of the lesson and remind students of what they’ve done, where they are, and where they are headed. Gather any feedback about the class from students (“muddiest points”, how to clarify, additional needs). Mention what will be covered in the next lesson and what assignments students should be working on between classes. Announce any upcoming tests, quizzes or projects. Math 101: Beginning Algebra Course Kit pg 9 See Unit 4 for example Developmental Studies Unit Overview - Teaching Insights & Strategies Unit Overview - Teaching Insights & Strategies Unit 1: Teaching Linear Equations Submitted by: Judy Walden & Mark Gollwitzer Purpose of Unit: This unit reviews simple arithmetic with signed numbers and solving linear equations and its applications. Where the Students are: Many of the students will be familiar with the concepts in this unit but don’t let them get complacent. The average student will still be having problems with fractions, and many of them don’t like doing math with the alphabet. This is a good time for them to perfect their skills. Where to Begin: Begin with a review of operations with real numbers, then bring in the initial concepts of linear equations. Connect to Previous Work: This unit is a review of skills they should have learned if they took MAT 031, MAT 032, or tested into MAT 101. Any review you can provide will be helpful, but it would be good to remind the students that these beginning sections may not be covered in detail. Students will be required to certify in the sections relevant to MAT 101. Challenges: Many students will want to work the problems in their head. We can usually cure them of this if we remind them that it is easy to get lost in problem solving. If they work the problems in their head the problems will eventually get so complicated they won’t be able to solve them. When this happens they will want to write something down but they won’t know where to begin. If on the other hand they start solving the simple problems on paper and writing out each step of the problem they will be building a strong foundation for their problem-solving practice and won’t notice as the problems get harder. Closure: Remind the students that math builds upon itself. Each time they learn new concepts, these concepts are building blocks for the next section and next chapter. They need to learn each set of concepts well so as to have a solid foundation to building upon. Math 101: Beginning Algebra Course Kit pg 10 Developmental Studies Lesson Plan Lesson 1a: Course Introduction & Basic Math Review Initial Lesson by: Bonnie Mullinix Course/Unit Focus: Beginning Algebra (Math 101), Solving Linear Equations Primary Course Outcome(s): 1. Perform arithmetic operations with real numbers and algebraic expressions. Learning Objectives: By the end of the session, learners will: 1.1. 1.2. Review the overview, structure and requirements of the course; Define and identify key terms: variable, term, constant, coefficient, and an algebraic expression; Practice simplifying expressions by combining like terms. 1.3. Materials: Whiteboard or interactive board notebook and pencil worksheet (provided) Activity Sheets & grid paper Duration: 1 class (75 min) Key Concepts & Skills: basic operations with real numbers order of operations operations with real numbers Time Description/Activity 10-20 min Introduction: Instructor introduces the students to the course, referencing the one-page summary and flowchart “Graphic Syllabus” for Math 101 to focus them on the math path they will be pursuing. Instructor points students to Blackboard, Text and software they will be using and demonstrates where to find the full syllabus and encourages them to print and review the syllabus in detail. 5-10 min Quick Kwiz Pt I: Instructor displays and distributes a quick quiz on real number operations (see attached), explaining that the purpose of this “kwiz” is to check and develop their understanding of basic math. So they will complete this alone and then share/compare their answers with students and self-assess their work and understanding. Students complete the quick kwiz individually. Math 101: Beginning Algebra Course Kit 15-25 min pg 11 Quick Kwiz Pt II: Students form groups of 3-4, introduce themselves and share and compare their answers, agreeing on the “right answer” within their group. Groups then ‘uncover’ the right answer using the IF-AT scratch-off answer sheets (or instructorled iClicker use). Instructor rotates among the groups and observes progress and patterns of understanding, answering questions where appropriate. Instructor asks students to score their kwiz in the following way: 1 pt for each individual correct answer ½ pt for each group correct answer next to each answer they now understand (from group work) ? next to each answer they don’t understand 5-10 min Kwiz Check: Instructor asks groups to identify all questions with a question mark next to them. Instructor uses these responses to focus on specific trouble problems, clarify steps, provide additional problems to try out, and demonstrate solutions. If needed for review: Instructor introduces acronyms to underscore appropriate order of operations – asking students to help identify what each stands for and ways to remember these orders: Complete the following acronyms: PEMDAS _______ _______ _______ _______ _______ _______ BEDMAS _______ _______ _______ _______ _______ _______ Hint: As they relate to the Order of Operations Write down how are they different? [2 min] Share/compare your thinking with a partner [2 min] Instructor invites students (or a student) to write their answers in the spaces as projected on the SmartBoard, and then projects the answers on top of what they have written and invites them to note the differences. 5-10 min 5-10 min If time allows, instructor provides additional problems (from text or software and/or on board) and students work in groups to solve them (to demonstrate and solidify their understanding of real number operations). Otherwise, problems may be assigned for homework. Summary – If necessary, instructor provides a brief lecturette to reinforce the key points of the lesson, address objectives/topics covered, assign homework and note focus of the next lesson. Homework Assignment: Log into Blackboard, Review syllabus (identify questions), Get text and complete the basic diagnostic in the online math software. [Back to Table of Contents] Assessment Strategies/Comments: Quick Quiz covering relevant problems. attached] Complete the following acronyms: [Need sample Math 101: Beginning Algebra Course Kit pg 12 PEMDAS Parentheses Exponents Multiply Divide Add Subtract BEDMAS Brackets Exponents Divide Add Subtract Remember by: Please Excuse My Dear Aunt Sally Big Elephants Destroy Mice And Snails Pink Elephants Destroy Mice And Snails Multiply Math 101: Beginning Algebra Course Kit pg 13 Review of Operations on Real Numbers Order of operations: When you have multiple operations, use the following priority: P – parentheses – Within grouping symbols, use order of operations. (i.e., grouping symbols also including brackets, absolute value, fraction bars) E – exponents MD – multiplication and division – Perform these left to right. AS – addition and subtraction – Perform these left to right. Many people remember this as: Please Excuse My Dear Aunt Sally. Example: 2 + 3(4 − 1 ∗ 7) 2 + 3(4 − 7) 2 + 3(−3) 2 + (−9) −7 Types of Numbers Natural numbers Whole numbers Integers Rational numbers Also called counting numbers Natural numbers plus zero Whole numbers plus negatives (typically, all tick marks on a number line) Numbers that can be written as a fraction of integers Note: “ratio” in rational numbers 1, 2, 3, 4, 5, … 0, 1, 2, 3, 4, 5, … …, -3, -2, -1, 0, 1, 2, 3, … All integers, plus fractions, terminating and repeating decimals 2 Irrational numbers Real numbers Numbers that cannot be written as a fraction of integers If written as a decimal, the digits never repeat. All numbers that are either rational or irrational. Ex: −4, 3, 1, 0, − , −.5, 3 .333333… Ex: √6, 𝜋 You can find all of these numbers on a real number line. Operations with Real Numbers Exponent Absolute value The number of times a number (base) is multiplied by itself Distance of a number from zero. 43 = 4 ∙ 4 ∙ 4 = 64 |2| = 2 |−4| = 4 −|−3| = −3 Adding Real Numbers With same signs With different signs Just add. Result has the same sign. Subtract the absolute values (ignore signs). Result has same sign as number −3 + (−9) = −12 −5 + 2 = −3 Math 101: Beginning Algebra Course Kit pg 14 −5 + 7 = 2 with largest absolute value. −(−𝑎) = 𝑎 −(−6) = 6 Same thing as adding the negative of the number. −5 − 9 = −5 + (−9) = −14 Double negative rule Subtracting Real Numbers −1 − (−2) = −1 + [−(−2)] = −1 + 2 = 1 Multiplying Real Numbers Times zero Same signs Opposite signs Dividing Real Numbers Zero −4 ∙ 0 = 0 −2 ∙ (−4) = 8 −3 ∙ 3 = −9 Any number times 0 = 0 Result is positive Result is negative Divide by 0: Result is “undefined” Divide 0 by non-zero: Result is zero. Helpful hint to remember which is which: 𝑁 𝑂 , 𝑂 −4 0 undefined 0 =0 −2 𝐾 Same signs Result is positive Opposite signs Result is negative −10 =5 −2 −12 = −3 4 25 = −5 −5 Properties of Real Numbers Commutative Property (addition, multiplication) Order doesn’t matter (−2) + 9 = 9 + (−2) (−2) ∙ 9 = 9 ∙ (−2) Associative Property (addition, multiplication) Identity Property (addition, multiplication) Inverse Property (addition, multiplication) Grouping doesn’t matter If you perform an operation using an identity, nothing changes −5 + 0 = −5 0 is additive identity 1 is multiplicative identity −9 ∙ 1 = −9 If you perform an operations using inverses (or opposites), you get the identity. −9 + 9 = 0 The multiplicative inverse has another name: reciprocal. Distributive Property [(−2) + 9] + (−1) = −2 + [9 + (−1)] When you have a number times a sum or difference, you can distribute. 1 −9 ∙ (− ) = 1 9 9(𝑎 + 5) = 9𝑎 + 45 −2(𝑏 − 7) = −2𝑏 + 14 −(𝑎 − 2) = −𝑎 + 2 Math 101: Beginning Algebra Course Kit pg 15 [Back to Table of Contents] Developmental Studies Lesson Plan Lesson 1b: Simplifying Algebraic Expressions Initial Lesson by: Habib Aghdami Course/Unit Focus: Beginning Algebra (Math 101), Solving Linear Equations Primary Course Outcome(s): 1. Perform arithmetic operations with real numbers and algebraic expressions. Learning Objectives: By the end of the session, learners will: 1.4. 1.5. 1.6. 1.7. Identify a variable, term, constant, coefficient, and an algebraic expression. Practice simplifying expressions by combining like terms. evaluate algebraic expressions translate phrases from words to algebraic expressions Emphasize: terminology how to identify like terms use of operations on negative numbers use of properties of real numbers how to use Hawkes Learning System and importance of homework Materials: Whiteboard or interactive board notebook and pencil worksheet (provided) Activity Sheets & grid paper Duration: 1 class (75 min) Time 5 min 5 min 5-10 min Description/Activity Intro – Instructor introduces session noting: This session will use Properties of Real Numbers and Operations with Real Numbers. It will also lead to Solving Linear Equations in the following sections. He will also note that today’s class will be taught using Hawkes in order to get the students proficient in the software. Future classes may also be taught using Hawkes. Instructor will open up the Hawkes lesson corresponding to Variable and Expressions. Then he will use the Instruct button to define each concept and show examples. Instructor should ask for one volunteer to do a couple of problems using the Practice button on Hawkes. The rest of the students should help come up with the answer Math 101: Beginning Algebra Course Kit Time 5 min 5-10 min 5 min 5-10 min 5 min 5-10 min pg 16 Description/Activity and the volunteer will key it in. The instructor should take the time to show the Tutor button and how it works. Instructor will open up the Hawkes lesson corresponding to Simplifying Expressions. Then he will use the Instruct button to define each concept and show examples. Instructor should ask for one volunteer to do a couple of problems using the Practice button on Hawkes (at least one problem with a squared variable and another with Distributive Property). The rest of the students should help come up with the answer and the volunteer will key it in. Instructor will open up the Hawkes lesson corresponding to Evaluating Expressions. Then he will use the Instruct button to define each concept and show examples. Instructor should ask for one volunteer to do a couple of problems using the Practice button on Hawkes (preferably problems with negative values for variables). Have the rest of the students work in small groups to come up with the answer and the volunteer will key it in. Instructor will open up the Hawkes lesson corresponding to Translating Phrases to Expressions. Then he will use the Instruct button to define each concept and show examples. Instructor should ask for one volunteer to do a couple of problems using the Practice button on Hawkes (preferably problems with negative values for variables). Have the rest of the students work in small groups to come up with the answer and the volunteer will key it in. Math 101: Beginning Algebra Course Kit pg 17 Simplifying Algebraic Expressions Activity Sheet Here are vocabulary terms used in algebra that are important to learn. Term: An algebraic term is either a number or a number multiplied by one or more variables. 52x2 - 9x + 36 = 7m + 82 Each green item is a separate term. Expression: An algebraic expression is made up of one or more algebraic terms. Expressions do not have equal signs. 52x2 - 9x + 36 = 7m + 82 Each blue item is a separate expression. Coefficient: A coefficient is the number part of a term with variables. 52x2 - 9x + 36 = 7m + 82 Each red item is a separate coefficient. Variable: In algebra, letters represent variables. 52x2 - 9x + 36 = 7m + 82 Each grey item is a separate variable. Constant: Terms with no variables are called constants. They are constants because their value is constant - it never changes. 52x2 - 9x + 36 = 7m + 82 Each purple item is a separate constant. Equation: An equation is a statement that two expressions are equal. An equation always has an equals sign ( = ). 52x2 - 9x + 36 = 7m + 82 3 + 5y = 8 Examples of equations Math 101: Beginning Algebra Course Kit pg 18 Simplify algebraic expressions by combining like terms. Like terms are terms with the same variable or variables. Like terms Unlike terms All constants are like terms. Constants and variables are not like terms. 2, -34, 0.59, ¼ 18, x All these terms have one variable, y. Terms with different variables are not like terms. 3y, -10y, y, 6y 7x, 7y All these terms have the same variable, t3. Terms with the same variable but different exponents are not like terms. t3, 9t3, ½t3, -14t3 10y, 3y2 All these terms have the same variable combination, xy. Terms with different variable combinations are not like terms. 29xy, 0.5xy, xy, -xy x, 8xy Example 1: One variable -x - 6 + 5x Simplified: -x and 5x are like terms, so combine them. -x - 6 + 5x = 4x - 6 The answer is 4x - 6 Example 2: Two variables 3y + 4 - y + 4x - 6 Simplified: Combine the x and y terms and the constants. 3y + 4 - y + 4x - 6 = 4x + (3y - y) + (4 - 6) = 4x + 2y + -2 = Math 101: Beginning Algebra Course Kit pg 19 This simplifies to 4x + 2y - 2 The answer is 4x + 2y - 2 Example 3: Simplify the expression and order your answer based on alphabetical letter and magnitude. 4x2 - 3y + x - 2x2 - 2 - 3y + 7 Simplified: Combine the terms and simplify. 4x2 - 3y + x - 2x2 - 2 - 3y + 7 = (4x2 - 2x2) + x + (-3y - 3y) + (-2 + 7) = 2x2 + x + (-6y) + 5 = This simplifies to 2x2 + x - 6y + 5 The answer is 2x2+ x - 6y + 5 [Back to Table of Contents] Math 101: Beginning Algebra Course Kit pg 20 Developmental Studies Lesson Plan Lesson 1c: Solving Linear Equations in One Variable Initial Lesson by: Judy Walden Course/Unit Focus: Beginning Algebra (MAT 101) / Linear Equations & Inequalities Primary Course Outcome(s): 2. Solve linear equations, inequalities, and formulas for specified variables. Learning Objectives: By the end of the lessons, learners will: 1.8. 1.9. 1.10. 1.11. 1.12. Identify linear equations in one variable. Use the addition and multiplication properties of equality to solve linear equations Solve more complicated linear equations that require simplication of expressions and both addition and multiplication properties of equality Solve linear equations that contain fractions by clearing fractions. Identify contradictions, identities, and conditional linear equations and state the solution set for each. Emphasize: The difference between solving an equation and simplifying an expression. What the solution of a linear equation means. The importance of writing down each step (i.e., not skipping steps, not solving equations in your head) The importance of checking your answer. Do not divide each side of an equation by a variable. The difference between how many solutions there are versus what the solutions are. Duration: 1.5 classes (115 min) Time 5 min 10 min 5-10 min Description/Activity Intro: Homework challenges discussed and/or collected. Instructor introduces lesson describing its importance, relationship to algebra and key concepts and skills to be addressed. Instructor opens up the Hawkes lesson corresponding to Solving Linear Equations: x+b=c and ax=c. Then uses the Instruct button to define each concept and show examples. Instructor should ask for one volunteer to do a couple of problems using the Practice button on Hawkes The rest of the students should work in small groups to help come up with the answer and the volunteer will key it in. The instructor will show the work on the board. Math 101: Beginning Algebra Course Kit Time 10 min 10-15 min 15 min 20-25 min 5-10 min pg 21 Description/Activity Instructor opens up the Hawkes lesson corresponding to Solving Linear Equations: ax+b=c. Then uses the Instruct button to define each concept and show examples. Instructor asks for one volunteer to do a couple of problems using the Practice button on Hawkes The rest of the students work in small groups to help come up with the answer and the volunteer will key it in. The instructor shows the work on the board. Instructor opens up the Hawkes lesson corresponding to Solving Linear Equations: ax+b=cx+d. Then uses the Instruct button to define each concept and show examples. Instructor asks for one volunteer to do a couple of problems using the Practice button on Hawkes The rest of the students should work in small groups to help come up with the answer and the volunteer will key it in. The instructor will show the work on the board. Summary - Instructor provides a brief lecturette to reinforce the key points of the lesson and foreshadow the next lesson and assigns homework (as needed) Assessment Strategies/Comments: Homework to be assigned in Tutorial software Activity: Algebra Balance Scales at the following link: http://nlvm.usu.edu/en/nav/category_g_4_t_2.html [Back to Table of Contents] Math 101: Beginning Algebra Course Kit pg 22 Developmental Studies Lesson Plan Initial Lesson by: Hala Nestberg Lesson 1d: Solving Absolute Value Equations Course/Unit Focus: Beginning Algebra (MAT 101) / Linear Equations & Inequalities Primary Course Outcome(s): 2. Solve linear equations, inequalities, and formulas for specified variables. Learning Objectives: By the end of the lessons, learners will: 1.13. 1.14. 1.15. 1.16. 1.17. 1.18. 1.19. Identify linear equations in one variable. Use the addition and multiplication properties of equality to solve linear equations Solve more complicated linear equations that require simplication of expressions and both addition and multiplication properties of equality Solve linear equations that contain fractions by clearing fractions. Identify contradictions, identities, and conditional linear equations and state the solution set for each. Solve absolute value linear equations. Graph the solution set for absolute value equations. Emphasize: The difference between solving an equation and simplifying an expression. What the solution of a linear equation means. The importance of writing down each step (i.e., not skipping steps, not solving equations in your head) The importance of checking your answer. Do not divide each side of an equation by a variable. The difference between how many solutions there are versus what the solutions are. The interpretation of the absolute value of a number as a distance from zero on a number line. Graphing on a number line the solution set for an absolute value equation. Duration: .5 classes (35 min) Time 5 min 10 min 10 min Description/Activity Transition: Instructor reviews what absolute value of a number means, finding the absolute value of a number, and evaluating absolute value expressions for a given value of a number Instructor opens up the Hawkes lesson corresponding to Solving Absolute Value (A.4) and uses the Instruct button to define each concept and show examples. Have students work on at least two practice examples. Math 101: Beginning Algebra Course Kit Time 5 min 5 min pg 23 Description/Activity Ask each student to team up with one other student. Have them check their answers with each other and explain how they came up with the solutions. Ask students to explain to each other the rationale. Summary - Instructor reiterates steps and rationale and that the � symbol and concept are useful in cases where how far values are from each other regardless of which is smaller than the other. Assessment Strategies/Comments: Homework to be assigned in Tutorial software [Back to Table of Contents] Math 101: Beginning Algebra Course Kit pg 24 Developmental Studies Lesson Plan Lesson 1e: Applications of Linear Equations Initial Lesson by: Judy Walden Course/Unit Focus: Beginning Algebra (MAT 101) / Linear Equations & Inequalities Primary Course Outcome(s): 2. Solve linear equations, inequalities, and formulas for specified variables. Learning Objectives: By the end of the lessons, learners will: 1.20. 1.21. Read and solve different kinds of applications using algebraic equations. Solve formulas for different variables Emphasize: Defining what the variable represents in each application problem. Using algebra to solve an application problem, not arithmetic. Solving formulas for different variables uses the same principles as solving linear equations. Duration: 2 class periods (2.5 hours) Time 5 min 15 min 20 min 15 min 20 min Description/Activity Intro: Homework challenges discussed and/or collected. Instructor introduces lesson describing its importance, relationship to algebra and key concepts and skills to be addressed. Instructor describes the steps to solving an application problem. Then selects a sample problem and goes thru these steps. Instructor goes to Practice button on Hawkes application section for number and consecutive integer problems, picks one sample of each type of problem, and works step by step to get the answer. Instructor finds another of one of these types of problems in Hawkes and has students work in groups to try to solve using algebra. After appropriate time, the instructor will review the problem. Instructor shows how to solve a formula for a different variable. Then goes to Hawkes Practice and have the students work in groups to solve. After appropriate time, the instructor will review the problem. Math 101: Beginning Algebra Course Kit Time 20 min 15 min 5-10 min pg 25 Description/Activity Instructor goes to Practice button on Hawkes application section for distance, interest, and average, picks one sample of each type of problem, and works step by step to get the answer. Instructor finds another of one of these types of problems in Hawkes and has students work in groups to try to solve using algebra. After appropriate time, the instructor will review the problem. Summary - Instructor provides a brief lecturette to reinforce the key points of the lesson and foreshadow the next lesson and assigns homework (as needed) Assessment Strategies/Comments: Homework to be assigned in Tutorial software [Back to Table of Contents] Math 101: Beginning Algebra Course Kit pg 26 Developmental Studies Unit Overview - Teaching Insights & Strategies Submitted by: Mark Gollwitzer Unit 2: Teaching Linear Inequalities & Graphing Linear Equations in Two Variables Purpose of Unit: In this unit we will be introducing the students to algebraic visualization, also known as graphing. This is probably the single most important unit in the development of mathematicians. Our goal is to explore the relationships between the equations, and their visual representations. If we do this properly the student should begin to understand how, as mathematicians we look at real world problems and find suitable equations that can produce solutions to those problems. If we give them a descent gimps into algebraic visualizations, they should understand the need to analyze and graph functions in future classes. Where the Students are: When the students get to this chapter they will be in relatively good spirits as it is still early in the semester, and they most of them will have done well on the first test. Unfortunately many of them will have passed the first test without putting in much time or effort. Although this unit is not that complicated as far as each individual skills they must master, we have found that this unit produces the lowest grades. What makes this unit hard is the fact that up until now the student has been given a skill or two they must master then tested on their ability to perform a specific task. Many of the students are solving problems by memorizing what to do, with little care for what they are actually looking for. This unit is different, as for every problem there may be several ways it can be worked and they will have to decide for themselves, how they want to attack the problem. Where to Begin: we need to spend some time introducing the concept of algebraic visualization to the students. This is simply done by having the students imagine, or actually drawing a picture as you describe it in words. It won’t take long for them to realize how challenging this is. If we then have them fold their paper in half both length and widthwise, it will be easy for them to see the grid, or axis they can measure from. We can then plot points then connect the dots. Next we take a simple linier equation say X + Y = 6 and have them find pairs of numbers that satisfy the equation. QED. Connect to Previous Work: when finding soultuions to linier equations we are using the problem solving skills they found in the previous chapters. Challenges: The greatest challenge in this unit is getting the students to understand before it is too late, that being able to perform the required skills, will not be sufficient they must actually make deeper connections. Math 101: Beginning Algebra Course Kit pg 27 Developmental Studies Lesson Plan Initial Lesson by: Hala Nestberg Lesson 2a: Solving Absolute Value Inequalities Course/Unit Focus: Beginning Algebra (MAT 101) / Linear Equations & Inequalities Primary Course Outcome(s): Learning Objectives: By the end of the lessons, learners will: 1. Solve simple linear inequalities involving absolute value. 2. Graph the solution set to absolute value inequalities using the real number line. Emphasize: The interpretation of the absolute value of a number as a distance from zero on a number line. Representing the solution set to an absolute value inequality by graphing the solution set on the real number line. Representing the solution set to an absolute value inequality by writing an appropriate inequality (union, intersection) Graphing on a number line the solution set for an absolute value equation. Duration: .5 classes (40 min) Time 5 min 15 min 15 min 5 min 5 min Description/Activity Transition: Instructor reviews what absolute value of a number means and what it means to solve inequalities. Instructor opens up the Hawkes lesson corresponding to Solving Absolute Value (A.4) and uses the Instruct button to define each concept and show examples. Have students work on at least four practice examples. Ask each student to team up with one other student. Have them use the loose paper activity to illustrate the solutions to absolute value inequalities and explain it to each other. Summary - Instructor reiterates steps and rationale. Assessment Strategies/Comments: Homework to be assigned in Tutorial software Activity: loose white paper; draw a number line and illustrate the less than a value versus the greater than the value. Activity: Algebra Balance Scales at the following link: http://nlvm.usu.edu/en/nav/category_g_4_t_2.html [Back to Table of Contents] Math 101: Beginning Algebra Course Kit pg 28 Developmental Studies Lesson Plan Initial Lesson by: Habib Aghdami Lesson 2b: Cartesian Coordinate System Course/Unit Focus: Beginning Algebra (MAT 101) / Linear Equations in Two Variables Primary Course Outcome(s): 3. Graph linear equations and determine the equation of a line. Learning Objectives: By the end of the session, learners will: 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. Plot points on a rectangular coordinate system. Identify what quadrant or axis a point lies on. Tell if an ordered pair is a solution of an equation in two variables or not. Complete an ordered pair that has one missing value. Read a bar graph. Read a line graph Materials: Rectangular Coordinate System and Reading Graphs, Activity/Discussion Sheets, Whiteboard, Computer & projector (or Overhead) Duration: 1 class (75 min) Time 20 min Description/Activity Intro: Instructor introduces lesson noting: this lesson contains properties of “Solving Linear Equations” (topics related to the previous chapter) and contains topics which will “Leadin” to the “Graphing Linear Equations in Two Variables”. Lecturette: Instructor provides a brief lecturette key points of the lesson and goes over relevant examples: 15-20 min - Sketches the Cartesian Plan on the board - Draws points in all quadrants (points out the characteristics of the points) - Draws point on axis (points out the characteristics of the points) - Evaluates a linear equation at several values for x and y - Goes over a small bar graph problem - Goes over a small line graph problem Instructor forms students in groups of 4-6 and provides each student a “Rectangular Coordinate Math 101: Beginning Algebra Course Kit Time pg 29 Description/Activity System and Reading Graphs Activity/Discussion Sheet” Students work individually to solve the problem. Students compare answers within their groups and discuss their challenges and findings 10-20 min 5-10 min Small Groups report out their solutions to the class. Instructor facilitates the discussion, posing questions, challenging groups/students to explain and clarify. Alternatively, instructor displays the Answer to Rectangular Coordinate System and Reading Graphs Activity/Discussion Sheet via a computer or an overhead. Summary - Instructor provides a brief lecturette to reinforce the key points of the lesson and foreshadow the next lesson. [Back to Table of Contents] Math 101: Beginning Algebra Course Kit pg 30 Graphing Linear Equations in Two Variables Activity/Discussion Sheet Name______________________________ Practice Problems 1a - 1b: Determine whether the equation is linear or not. 1a. y = 2x - 1 1b. Practice Problems 2a - 2b: Graph the linear equation. 2a. y = 2x - 1 2b. Math 101: Beginning Algebra Course Kit Answers to Graphing Linear Equations in Two Variables Activity/Discussion Sheet Answer/Discussion to 1a y = 2x - 1 If we subtract 2x from both sides, then we can write the given equation as -2x + y = -1. Since we can write it in the standard form, Ax + By = C, then we have a linear equation. Answer/Discussion to 1b If we add x squared to both sides we would end up with . Is this a linear equation? Note how we have an x squared as opposed to x to the one power. It looks like we cannot write it in the form Ax + By = C, because the x has to be to the one power, not squared. So this is not a linear equation. Answer/Discussion to 2a y = 2x - 1 Step 1: Find three ordered pair solutions. The three x values I'm going to use are -1, 0, and 1. (Note that you can pick ANY three x values that you want. You do not have to use the values that I picked.) You want to keep it as simple as possible. The following is the chart I ended up with after plugging in the values I mentioned for x. pg 31 Math 101: Beginning Algebra Course Kit pg 32 x y = 2x - 1 (x, y) -1 y = 2(-1) - 1 = -3 (-1, -3) 0 y = 2(0) - 1 = -1 (0, -1) 1 y = 2(1) - 1 = 1 (1, 1) Step 2: Plot the points found in step 1. Step 3: Draw the graph. Answer/Discussion to 2b Math 101: Beginning Algebra Course Kit pg 33 Step 1: Find three ordered pair solutions. The three x values I'm going to use are -1, 0, and 1. (Note that you can pick ANY three x values that you want. You do not have to use the values that I picked.) You want to keep it as simple as possible. The following is the chart I ended up with after plugging in the values I mentioned for x. x y = -1/2x (x, y) -1 y = -1/2(-1) = 1/2 y = -1/2(0) = 0 y = -1/2(1) = -1/2 (-1, 1/2) 0 1 Step 2: Plot the points found in step 1. Step 3: Draw the graph. (0, 0) (1, -1/2) Math 101: Beginning Algebra Course Kit pg 34 Math 101: Beginning Algebra Course Kit pg 35 Developmental Studies Lesson Plan Lesson 3b: Exploring Linear Data Initial Lesson by: Habib Aghdami Course/Unit Focus: Beginning Algebra (MAT 101) / Graphing Primary Course Outcome(s): 3. Graph linear equations and determine the equation of a line. Learning Objectives: By the end of the session, learners will: 2.7. 2.8. 2.9. 2.10. Explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope. Use graphs to analyze the nature of changes in quantities in linear relationships. Describe the importance of scatterplots and use them to display data. Estimate and write equations of lines. Materials: Grid Paper (several sheets per student or group) Bike Weights and Jump Heights Activity Sheet Weights and Drug Doses Activity Sheet, an overhead or a computer Duration: 2 class periods (2.5 hrs) Time 5 min 15-20 min Description/Activity Intro – Instructor introduces lesson noting: this lesson contains properties of “Solving Linear Equations” (topics related to the previous chapter) and contains topics which will “Leadin” to the “Solving the System of Linear Equations” (topics related to the next chapter). Instructor invites student to choose which problem they wish to work on (Bike Jump or Medicine Doses). Students then form groups of 4-6 for each of the two problems. Students work individually to solve the problem Students compare answers within their groups and discuss their findings. 20-30 min Groups report out their solutions to the class. Instructor facilitates the discussion, posing questions, challenging groups/students to explain and clarify. 20-30 min Students find a partner who did the other problem and meet and share results; each talking the other through their problems. 5-10 min Instructor provides a brief lecturette to reinforce the key points of the lesson and foreshadow the next lesson. Math 101: Beginning Algebra Course Kit pg 36 [Back to Table of Contents] Math 101: Beginning Algebra Course Kit pg 37 Developmental Studies Lesson Plan Initial Lesson by: Habib Aghdami Lesson 2c: Graphing Linear Equations in Two Variables Course/Unit Focus: Beginning Algebra (MAT 101) / Graphing Primary Course Outcome(s): 3. Graph linear equations and determine the equation of a line. Learning Objectives: By the end of the session, learners will: 2.11. 2.12. 2.13. 2.14. 2.15. Recognize when an equation in two variables is a linear equation (or not); Graph a linear equation; Find the x- and y-intercepts of a linear function; Graph a linear function using the x- and y-intercepts; Graph vertical and horizontal lines. Materials: Graphing Linear Equations in Two Variables Activity/Discussion Sheet, a Computer or an Overhead Duration: 1 class period (75 min) Time 15 min Description/Activity Intro: Instructor introduces lesson noting: this lesson contains properties of “Rectangular Coordinate System and Reading Graphs” (topics related to the previous section) and contains topics which will “Leadin” to the “Slope of a Line” (topics related to the next section). Lecturette: Instructor provides a brief lecturette key points of the lesson and goes over relevant examples: 15 -20 min - Characteristics of a linear equations vs. a non linear equation - Evaluating a linear equations in two variables - Find the x- and y-intercepts of a linear function. - Graph a linear function using the x- and y-intercepts. - Graph vertical and horizontal lines. Instructor forms students in of 4-6 and provides each student a “Graphing Linear Equations in Two Variables Activity/Discussion Sheet” and “Intercept Activity/Discussion Sheet” Students work individually to solve the problem Math 101: Beginning Algebra Course Kit Time pg 38 Description/Activity Students compare answers within their groups and discuss their challenges and findings 10-20 min 5-10 min Groups report out their solutions to the class. Instructor facilitates the discussion, posing questions, challenging groups/students to explain and clarify. Alternatively, instructor displays the Answer to Graphing Linear Equations in Two Variables Activity/Discussion Sheet and “Intercept Activity/Discussion Sheet via a computer or an overhead Instructor provides a brief lecturette to reinforce the key points of the lesson and foreshadow the next lesson. [Back to Table of Contents] Math 101: Beginning Algebra Course Kit pg 39 Developmental Studies Lesson Plan Lesson 2d: Graphing: Slope-Intercept Form Initial Lesson by: Habib Aghdami Course/Unit Focus: Beginning Algebra (MAT 101) / Graphing Primary Course Outcome(s): 3. Graph linear equations and determine the equation of a line. Learning Objectives: By the end of the session, learners will: 2.16. 2.17. 2.18. Find the slope given a graph, two points or an equation. Write a linear equation in slope/intercept form. Determine if two lines are parallel, perpendicular, or neither. Materials: Slope Activity/Discussion Sheet, a Computer or an Overhead Duration: 1 class period (75 min) Time 20 min Description/Activity Intro: Instructor introduces lesson noting: this session contains properties of “Graphing Linear Equations in Two Variables” (topics related to the previous section) and contains topics which will “Leadin” to the “Equations of Lines” (topics related to the next section). Lecturette: Instructor provides a brief lecturette key points of the lesson and goes over relevant examples: 15-20 min - Rise/Run Formula - Slope – Intercept Formula - Point – Slope Formula - Characteristics of “X = c” Equation - Characteristics of “Y = c” Equation Instructor forms students in of 4-6 and provides each student a “Slope Activity/Discussion Sheet” Students work individually to solve the problem Students compare answers within their groups and discuss their challenges and findings Math 101: Beginning Algebra Course Kit 10-20 min 5-10 min pg 40 Groups report out their solutions to the class. Instructor facilitates the discussion, posing questions, challenging groups/students to explain and clarify. Alternatively, instructor displays the Answer to Slope Activity/Discussion Sheet via a computer or an overhead Instructor provides a brief lecturette to reinforce the key points of the lesson and foreshadow the next lesson. [Back to Table of Contents] Math 101: Beginning Algebra Course Kit pg 41 Developmental Studies Lesson Plan Lesson 2e: Graphing: Point-Slope Form Initial Lesson by: Habib Aghdami Course/Unit Focus: Beginning Algebra (MAT 101) / Graphing Primary Course Outcome(s): 3. Graph linear equations and determine the equation of a line. Learning Objectives: By the end of the session, learners will: 2.19. 2.20. 2.21. Use the slope/intercept form to write a linear equation given the slope and y-intercept. Use the slope/intercept form to graph a linear equation. Use the point/slope equation to set up an equation given: a. any point on the line and the slope. b. two points on the line. c. a point on the line and a parallel line. d. a point on the line and a perpendicular line. Materials: Slope Activity/Discussion Sheet, a Computer or an Overhead Duration: 1 class period (75 min) Time 20 min Description/Activity Intro: Instructor introduces lesson noting: this lesson contains properties of “Slope of Lines” (topics related to the previous section) and contains topics which will “Leadin” to the “Applications of Two Variable. Lecturette: Instructor provides a brief lecturette key points of the lesson and goes over relevant examples: 15-00 min Rise/Run Formula Slope – Intercept Formula Point – Slope Formula Characteristics of “X = c” Equation Characteristics of “Y = c” Equation Characteristics of parallel lines Characteristics of perpendicular lines Instructor forms students in of 4-6 and provides each student a “Equations of Lines Activity/Discussion Sheet” Students work individually to solve the problem Students compare answers within their groups and discuss their challenges and findings Math 101: Beginning Algebra Course Kit Time 10-20 min 5-10 min pg 42 Description/Activity Groups report out their solutions to the class. Instructor facilitates the discussion, posing questions, challenging groups/students to explain and clarify. Alternatively, instructor displays the Answer to Equations of Lines Activity/Discussion Sheet via a computer or an overhead Instructor provides a brief lecturette to reinforce the key points of the lesson and foreshadow the next lesson. [Back to Table of Contents] Math 101: Beginning Algebra Course Kit pg 43 Developmental Studies Lesson Plan Lesson 2f: Applications of Two Variable Equations Initial Lesson by: Habib Aghdami Course/Unit Focus: Beginning Algebra (MAT 101) / Graphing Primary Course Outcome(s): 3. Graph linear equations and determine the equation of a line. Learning Objectives: By the end of the session, learners will: 2.22. … 2.23. … 2.24. … Materials: Duration: 1 class period (75 min) Time 20 min Description/Activity Intro: Instructor introduces lesson noting: this lesson contains properties of “Slope of Lines” (topics related to the previous section) and contains topics which will “Leadin” to the “Applications of Two Variable. Lecturette: Instructor provides a brief lecturette key points of the lesson and goes over relevant examples: 5-10 min 15-25 min 10-20 min 5-10 min Instructor provides a brief lecturette to reinforce the key points of the lesson and foreshadow the next lesson. [Back to Table of Contents] Math 101: Beginning Algebra Course Kit pg 44 Developmental Studies Unit Overview - Teaching Insights & Strategies Submitted by: Mark Gollwitzer Unit 3: Teaching Systems of Linear Equations Purpose of Unit: in this unit we will be teaching students the substitution, and addition methods for solving problems with more than one variable. Where the Students are: Many of the students just received their first bad test grade in math since high school. Because of this many of them will be questioning whether they can succeed in math at all. This chapter separates those that are willing to work hard for a grade from the rest. If handled correctly this is a wonderful chapter for helping the students develop mathematical maturity. Where to Begin: We need to convince the students that the material in this chapter is within their grasp. We can do this by explaining to them that they will only be asked to learn two new skills, i.e. the substitution, and addition methods. We then start with the substitution method as they have already been doing this when solving word problems. This method should not be too far out of their reach, therefore they should gain some confidence. When we show them the addition method they will grasp it enthusiastically, for after working with the substitution method for a couple of days the addition method will seem like childes play. Then all we need do is motivate them to practice, practice, and practice. Connect to Previous Work: The chapter starts out solving systems by graphing. This is a review of the last chapter. Teaching substitution is easily done by reminding them of how they replaced unknowns, with expressions when solving word problems. The addition method can easily be taught by reminding the students about the addition property of equality. Challenges: Remember, the students are usually more than a little frustrated with their performance on the last test, and beginning to doubt themselves. Because of this we need to expose the students to as many different levels of difficulty as possible, so that when they see the problems on their homework and tests they don’t panic. Closure: This is a good chapter for pushing students to work hard. The best thing about this chapter is, the problems are complicated and time consuming enough, so that when their test grades improve they will see the value in homework, and begin to gain some of the confidence they will need in future chapters. Math 101: Beginning Algebra Course Kit pg 45 Developmental Studies Lesson Plan Lesson 3a: Solving Systems of Equations by Graphing Initial Session by: W. Parker Course/Unit Focus: MAT 101/Unit 3: Systems of Linear Equations Primary Course Outcome(s): 4. Solve systems of two linear equations and interpret solutions; Learning Objectives: By the end of the lesson, learners will: 3.1. Solve a system of equations by using appropriate graphing techniques. 3.2. Identify the solution of the system 3.3. Interpret the significance of the solution Materials: Graph paper and straight edge/ruler Duration: 75 minutes Time Description/Activity 5 min. Review objectives. Introduce topic and uses in mathematics. 10 min Define system of equations, the possible solutions, and how to interpret solutions. 10 min 20-25min 20 min Review and demonstrate graphing of linear equations by different methods: Table of values; slope/intercept. Demonstrate solving systems of equations by graphing. Show examples with: one solution, infinite solutions, and no solutions. Review how a solution is checked. Activity: Small group-solving practice problems. Students work individually and in groups to solve 5 problems (either even or odd by group). Groups check their answers with each other (10 min). One person from the even group pairs with a person from the odd group and shares answers (5-10 min). 5-10 min Review solutions and interpretation of solutions. Instructor encourages students to try and complete any problems that they did not attempt in class for homework in addition to completing certification for Hawkes 10.1. Assessment Strategies/Comments: Hawkes Learning Systems 10.1 [Back to Table of Contents] Math 101: Beginning Algebra Course Kit pg 46 Math 101: Beginning Algebra Course Kit pg 47 Math 101: Beginning Algebra Course Kit pg 48 [Back to Table of Contents] Math 101: Beginning Algebra Course Kit pg 49 Developmental Studies Lesson Plan Lesson 3b-1: Solving System of Equations by Substitution Initial Session by: W. Parker Course/Unit Focus: MAT 101/ Systems of Linear Equations Primary Course Outcome(s): 4. Solve systems of equations and interpret solutions; Learning Objectives: By the end of the session, learners will: 3.4. Use the substitution method to find the solution of the system of equations. 3.5. Identify the solution of the system 3.6. Interpret the significance of the solution Materials: Text Duration: 40 minutes Time 5 min. Description/Activity Review objectives. Introduce topic and uses in mathematics. 5 min Review and demonstrate solving a system of equations by graphing 10 min Demonstrate solving systems of equations by substitution. Review how a solution is checked and interpreted. 15min Activity: Small group-solving practice problems. Students work cooperatively to solve 3 problems from worksheet. 5min Review solutions by substitution and interpretation of solutions. Student volunteer will explain solution from the group. Additional problems from worksheet and certification for Hawkes 10.2 assigned for homework. Assessment Strategies/Comments: Hawkes Learning Systems 10.2 [Back to Table of Contents] Math 101: Beginning Algebra Course Kit pg 50 Math 101: Beginning Algebra Course Kit pg 51 [Back to Table of Contents] Math 101: Beginning Algebra Course Kit pg 52 Math 101: Beginning Algebra Course Kit pg 53 Developmental Studies Lesson Plan Lesson 3b-2: System of Equations by Addition Initial Session by: W. Parker Course/Unit Focus: MAT 101/Unit 3: Systems of Linear Equations Primary Course Outcome(s): 4. Solve systems of equations and interpret solutions; Learning Objectives: By the end of the session, learners will: 3.7. Use the addition method to find the solution of the system of equations. 3.8. Identify the solution of the system 3.9. Interpret the significance of the solution Materials: Text Duration: 35 minutes Time 2 min Description/Activity Review objectives. Introduce topic and compare addition method substitution method. 5 min Review substitution to solve a system of equations, the possible solutions, and how solutions are interpreted. 5min Demonstrate solving the same system of equations by addition method. Review how the solution is checked. 5min Demonstrate solving different systems of equations by addition method. Show examples with one solution, infinite solutions, and no solutions. Review how a solution is checked and interpretation of the solution. 15min Activity: Small group-solving practice problems. Students work in groups to solve 2-3 problems from worksheet. Groups will report/discuss using substitution and addition methods for solving systems of equations. 3 min Review solutions and interpretation of solutions. Instructor encourages students to complete problems from the worksheet in addition to completing certification for Hawkes 10.3 Assessment Strategies/Comments: Hawkes Learning System 10.3 Math 101: Beginning Algebra Course Kit pg 54 Math 101: Beginning Algebra Course Kit pg 55 [Back to Table of Contents] Math 101: Beginning Algebra Course Kit pg 56 Developmental Studies Lesson Plan Lesson 3c: Applications of Linear Systems Initial Session by: W. Parker Course/Unit Focus: MAT 101/Unit 3: Systems of Linear Equations Primary Course Outcome(s): 4. Solve systems of equations and interpret solutions; Learning Objectives: By the end of the session, learners will: 3.10. 3.11. 3.12. 3.13. Convert word problems to systems of equations Use an appropriate method to find the solution of the system of equations. Identify the solution of the system Interpret the significance of the solution Materials: Text Duration: 75 minutes Time 2 min 25 min Description/Activity Review objectives. Activity: Small group-solving practice problems. Students work in groups to solve 2-3 problems from worksheet. Groups will identify: 1. 2. 3. 4. 5. Variables/unknowns Two equations Method used to find solution Solution Interpretation of solution Groups will report/discuss the methods used for solving the problem by systems of equations. 25min Demonstrate converting and solving word problems using systems of equations. Explain why a system of equations might be necessary or more appropriate than an equation with a single variable. Review how the solution is checked in the original word problem. 20min Activity: Small group-solving practice problems. Students work in different groups to solve 2-3 problems from worksheet. Groups will report/discuss using methods for solving systems of equations. Instructor will assist groups as necessary. 3 min Review solutions and interpretation of solutions. Instructor encourages students to complete problems from the worksheet in addition to completing certification for Hawkes 10.4 Assessment Strategies/Comments: Hawkes Learning System 10.4 Math 101: Beginning Algebra Course Kit pg 57 Math 101: Beginning Algebra Course Kit pg 58 Math 101: Beginning Algebra Course Kit [Back to Table of Contents] pg 59 Math 101: Beginning Algebra Course Kit pg 60 Developmental Studies Lesson Plan Lesson 3d: Exponents Initial Lesson by: Mark Gollwitzer Course/Unit Focus: Beginning Algebra (MAT 101) / Systems of Linear Equations - Exponents Primary Course Outcome(s): Use the rules of exponents to simplify algebraic expressions. Learning Objectives: By the end of the session, learners will: 3.14. Build and use the rules for exponents; 3.15. Practice and apply rules for exponents; 3.16. Structure and solve problems with scientific notation. Materials: Smartboard/Whiteboard; preselected problems, quiz. Duration: 2 class sessions/2.5 hours Key Concepts & Skills: Time 15 min Description/Activity Explain the definition of exponent: 23 = 2 ∙ 2 ∙ 2 , 𝑎𝑚 = 𝑎 ∙ 𝑎 ∙ 𝑎 … ∙ 𝑎. Show the students how to expand and discover product and quotient rule for 𝑎2 ∙ 𝑎3 , and 30-45 min 𝑎5 𝑎3 𝑎 3 𝑎2 𝑏 𝑏 . 2 Have the students break up into groups and develop the rules for (𝑎𝑏)3 , (𝑎2 𝑏 3 )2 , ( ) , ( 3) , 𝑎5 𝑎7 , and 𝑎2 𝑎2 50-80 min Have students use exponent rules in class to simplify selected problems making students aware of the potentially confusing ways the problem could be presented. 15-40 min Explain what a polynomial is, then show the students how to divide a polynomial by a monomial. Two or three examples are enough. 10-15 min Give the students a short 3-5 question quiz on division of a polynomial by a monomial. Note: Assessment. The quiz should contain 1 monomial/monomial, 2 binomial/monomial, and 3 trinomial/monomial. [Back to Table of Contents] Assessment Strategies/Comments: Practice problems & quiz Math 101: Beginning Algebra Course Kit pg 61 to be integrated w/ Lesson 3d Exponents above Initial Lesson by: Mark Gollwitzer Lesson 3d-2: Application of Exponents - Scientific Notation Original Lesson 4d Course/Unit Focus: Beginning Algebra (MAT 101) / Graphing Primary Course Outcome(s): 4. Simplify, evaluate, and perform arithmetic operations with polynomials. & 5.Use the rules of exponents to simplify algebraic expressions. Learning Objectives: By the end of the session, learners will: 3.17. 3.18. 3.19. Convert numbers to scientific notation. Convert numbers from scientific notation. Multiply and divide numbers in scientific notation. Materials: Smart board /white board (Computer lab, where needed and other materials as required) Duration: 30 min Time 5 min 3 min Description/Activity Ask the students if they know what the national debt is. Talk about the national debt clock by showing or explaining that the smaller numbers are changing so fast they aren’t worth mentioning. Explain that as this is a large number it would be nice if we could express it in a more compact way. Of to the side explain 5x10= 50, 5x100 or 10^=500, and 5x1000 or 10^3 =5000. Use this new understanding to convert national debt to 13.5x10^12 Help the student see the relationship between the number of zero’s and the exponent. 2min Explain that while it is easier to understand $13.5 Trillion in S.I. we would have moved the decimal one more place, making it 1.35x10^13 to fit the true form of S.I... Note: This helps them see that there is a specific form for scientific notation. 5 min Discuss the number of people in the United States. Turn this number into S.I. 5 min Ask the students how much every American citizen owes on the national debt. Show them how to use the rules of exponents to do the division. 5 min Ask the student how much an eyelash weighs (0.000304grams). Show the student how to convert extremely small numbers to S.I.. Have students convert this weight to pounds. Have them express this number in standard form. 5 min Ask the students what the total weight of all eyelashes in America. Assume 100-150 eyelashes per person. [Back to Table of Contents] Math 101: Beginning Algebra Course Kit pg 62 Developmental Studies Unit Overview - Teaching Insights & Strategies Unit 4: Teaching Polynomials and Factoring Submitted by: Mark Gollwitzer & Debbie Moran Purpose of Unit: To understand the problems inherent in teaching exponents and polynomials to the developmental student. Where the Students are: The material in this section is relatively easy and accessible for most students. When the students get this far many of them have struggled through, and in many cases been disappointed with their grades on the chapters dealing with the graphing of linear equations. The last day to withdraw has just passed so some students will have withdrawn from the class while, those that remain may have grades that are borderline. In the following lesson plans we are trying to teach in a way that gives the student some confidence back. Where to Begin: Chapter 11 is already in progress and working with exponents has been a good experience. They find this topic to be a bit easier than the graphing and systems they have been studying previously. Those students that need to pick their grades up will get motivated now. Connect to Previous Work: At this point students have been working with polynomials, but just the simplest type while learning to solve equations. Once this idea is pointed out, the student will pick up the idea of simplifying polynomials quickly. Function notation was introduced in chapter 9 and will be reinforced here to evaluate polynomial expressions. Challenges: Although long division of polynomials may look difficult at first glance, if we carefully compare it to long division of numbers, they usually can get it as their confidence by this point is coming back. Closure: Students seem to thrive more on totally algebraic manipulation and this chapter provides just that. Their grades should recover nicely and they will be in a better situation for the sections on factoring. Math 101: Beginning Algebra Course Kit pg 63 Developmental Studies Lesson Plan Lesson 4a: Introduction to Polynomials & Operations Initial Lesson by: Debbie Moran Course/Unit Focus: Beginning Algebra (MAT 101) / Polynomials Primary Course Outcome(s): 4. Simplify, evaluate, and perform operations with polynomials and 5. Factor polynomials and solve quadratic equations by factoring. Learning Objectives: By the end of the session, learners will: 4.1. Define a polynomial, and learn how to classify polynomials. 4.2. Evaluate a polynomial for given values of the variable. 4.3. Be able to add and subtract polynomials. 4.4. Simplify algebraic expressions by removing grouping symbols and combining like terms. Materials: Duration: 1 class period (75 min) Time 10-15 min Description/Activity Entry Activity: Instructor projects entry problem(s) (or posts on whiteboard) for learners to work on as they enter. [from 9.5] 𝑓(𝑥) = 𝑥 2 − 3𝑥 + 2, find 𝑓(4), 𝑓(0), 𝑓(−3) Simplify: [from 7.7] 4(𝑦 + 3) − 5(𝑦 − 2) Simplify: [from 7.7] 2𝑥 2 − 2𝑦 + 5𝑥 2 + 6𝑥 2 Answer questions from previous class period. 40 min Section 11.3 Give students the definition for a Monomial, Polynomial found on page 875. Introduce the special names for 1st -, 2nd - and 3rd -degree polynomials and give examples. Evaluate polynomials using the function notation. Section 11.4 This should be an easy section as the student already have experienced adding and subtracting like terms as well as removing parenthesis with the distributive property. Emphasis that subtraction of two polynomials must remove the parenthesis by distributing a -1 factor to all terms within the parenthesis. Subtracting in a vertical format is important for long division later in the chapter. 5-10 min Instructor provides a brief lecturette to reinforce the key points of the lesson and foreshadow the next lesson. Current assignments/homework is mentioned. [Back to Table of Contents] Assessment Strategies/Comments: Hawkes Learning software certification. Math 101: Beginning Algebra Course Kit pg 64 Developmental Studies Lesson Plan Lesson 4b: Operations on Polynomials – Multiplying Initial Lesson by: Mark Gollwitzer & Debbie Moran Course/Unit Focus: Beginning Algebra (MAT 101) / Polynomials Primary Course Outcome(s): 4. Simplify, evaluate, and perform arithmetic operations with polynomials. Learning Objectives: By the end of the session, learners will: 4.5 Multiply monomials by polynomials 4.6 Multiply two polynomials 4.7 Multiply polynomials using the FOIL method 4.8 Multiply binomials find products that are the difference of squares 4.9 Multiply binomials find products that are perfect square trinomials. Materials: Whiteboard/Chalk board Duration: 1.5 hours Time 5-15 min Description/Activity Multiplying Polynomials - Remind the students how the distributive property works. Expand their understanding by showing examples of progressively more complicated monomials distributed into longer and more complicated polynomials and demonstrating the use of rules for exponents to multiply polynomials. 10-20 min Multiplying Binomials & Trinomials - Show the students how to use the distributive property to multiply binomials by trinomials and trinomials by trinomials. Notes: Clarify how distributive property compares to FOIL; Highlight how when collecting like terms we add coefficients not exponents. 20-30 min Special product rules - Introduce the mnemonic FOIL to perform the operation of multiplying two binomials. This mnemonic will remind students that there are 4 products that must be found when multiplying two binomials. Expose the students to the special product rules and short cuts for multiplying polynomials. 5 min Instructor provides a brief lecturette to reinforce the key points of the lesson and foreshadow the next lesson. Current assignments/homework is mentioned. [Back to Table of Contents] Assessment Strategies/Comments: Hawkes Learning software certification. Math 101: Beginning Algebra Course Kit pg 65 Developmental Studies Lesson Plan Lesson 4c: Opera tions on Polynomia ls Division Initial Lesson by: Mark Gollwitzer & Debbie Moran Course/Unit Focus: Beginning Algebra (MAT 101) / Polynomials Primary Course Outcome(s): 4. Simplify, evaluate, and perform arithmetic operations with polynomials. Learning Objectives: By the end of the session, learners will: 4.10 4.11 4.12 4.13 Divide a polynomial by polynomials by a monomial. Divide polynomials using the long division algorithm. Check a long division using by multiplying quotient by divisor and adding the remainder. Use place holders for missing terms. Materials: Smart board/ white board / PowerPoint (Computer lab, where needed and other materials as required) Duration: 70 min Time 10min Description/Activity Show the students how we divide a polynomial by a monomial, 18𝑥 5 −6𝑥 4 +9𝑥 3𝑥 = 18𝑥 5 3𝑥 − 6𝑥 4 3𝑥 + 9𝑥 3𝑥 and then reduce each fraction. 25 min Show the students another method using an example of a polynomial divided by a larger polynomial and explain the need for a new method i.e. can’t cancel terms only factors. (Remind the student how not to cancel.) example: 𝑥 2 −2𝑥−20 𝑥+4 Review long division of numbers by hand. 695/31. Explain in detail and show the math for all the steps. Note when thinking about how many times 31 goes into 69 we can ignore the 1 in 31 Note: Use of long division; also, many Hispanic students didn’t learn the galley method of long division. Show example of long division. Take 𝑥 2 +5𝑥+6 𝑥+3 and as you are doing the long division note we can Math 101: Beginning Algebra Course Kit pg 66 Time Description/Activity ignore the +3 in x+3 just like we ignored the 1 in 31. Note that we draw the line and change the signs in order to do the subtraction. Use distributive property to check. Have students do one. 5 min Show a new example. Take 𝑥 2 +7𝑥−12 𝑥−3 and as you are doing the long division note we can ignore the -3 in x-3 just like we ignored the 1 in 31. Note that we draw the line and change the signs in order to do the subtraction. This time make sure they understand when we subtract a negative we actually add. Hence draw the line and change the signs. Have students do one. 5 min Using this example 4𝑥 2 +12𝑥+9 2𝑥+3 note we can take 4𝑥 2 2𝑥 to get started. Then have students do one. Note: Many students have trouble figuring out what they need to multiply by. 20 min Give the students problems with 3rd and 4th degree polynomials, some that have remainders, and some with missing terms. While students are working on them explain the need for place holders. Note: help the students make the final connections. [Back to Table of Contents] Assessment Strategies/Comments: Hawkes Learning software certification. Math 101: Beginning Algebra Course Kit pg 67 Developmental Studies Lesson Plan Initial Lesson by: Debbie Moran Lesson 4d: The Greatest Common Factor, Factor by Grouping Course/Unit Focus: Beginning Algebra (MAT 101) / Factoring Primary Course Outcome(s): 5. Factor polynomials and solve quadratic equations by factoring. Learning Objectives: By the end of the session, learners will: 4.10 Find the GCF of a set of terms 4.11 Factor polynomials by factoring out the monomial GCF. 4.12 Factor polynomials by grouping Duration: 90 min Time 10 min 30 min Description/Activity Begin by pointing out the difference between “factors” and “product”. (15)(3) = 45 Factors are 15 and 3 and the product is 45. Talk about factorization of numbers. Find the GCF of two larger numbers, 30 and 54 using the same method as with polynomials. Continue with an example of monomials, 150𝑥𝑦, 250𝑦 2 𝑥 3 , 100𝑥 2 𝑦 2 Practice finding and factoring out the GCF with more than one term. Start by using the same terms as in the previous example: 150𝑥𝑦 + 250𝑦 2 𝑥 3 − 100𝑥 2 𝑦 2. Use the GCF and rewrite as a product of the GCF and the remaining factors: 50𝑥𝑦(3 + 5𝑦𝑥 2 − 2𝑥𝑦) Be sure to point that when a whole term is factored out, a 1 (or -1) must be used to retain the “place” in the polynomial. This way when the GCF is re-distributed to each term, the original polynomial will be the result. 10 min 30 min Allow students to practice with several problems. Factoring by Grouping – a development idea: ax + ay = a(x+y) $x + $y= $(x+y) (Trash)x + (Trash)y = (Trash)(x+y) “Taking out the Trash” (a + b) x + (a+b)y = (a+b)(x+y) Now proceed using polynomial examples. Be sure to include examples when a -1 is the GCF for the 2nd group of terms. Also show students that the terms can be reordered and the same resulting factorization. Math 101: Beginning Algebra Course Kit Time pg 68 Description/Activity Do not allow students to group by using parenthesis. The expression 𝑥𝑦 + 5𝑥 + 3𝑦 + 15 usually becomes (𝑥𝑦 + 5𝑥)(+3𝑦 + 15) showing these terms now as a product. It might be best to use a squiggly underline for each pair of terms or a brace . Allow students to practice several of these grouping problems. 5-10 min Instructor provides a brief lecturette to reinforce the key points of the lesson and foreshadow the next lesson. [Back to Table of Contents] Assessment Strategies/Comments: Hawkes Learning software certification. Math 101: Beginning Algebra Course Kit pg 69 Developmental Studies Lesson Plan Initial Lesson by: Debbie Moran Lesson 4e: Factoring Trinomials Course/Unit Focus: Beginning Algebra (MAT 101) /Factoring Primary Course Outcome(s): 5. Factor polynomials Learning Objectives: By the end of the session, learners will: 4.14 Factor trinomials with a coefficient of 1 4.15 Factor trinomials with a coefficient of 1 after factoring out the GCF. Duration: 50 min. Time 15 min 15 min Description/Activity Introduce Factoring an equation of the form 𝒙𝟐 + 𝒃𝒙 + 𝒄 by showing the product of two binomials and how the numerical portions are related. (𝒙 + 𝟐)(𝒙 + 𝟓) = 𝒙𝟐 + 𝟕𝒙 + 𝟏𝟎 where 7 = 2+5 and 10 = (2)(5) Now, show the reverse of this problem looking for factors of 10 that add to be 7. Include other examples with values for b and c being different combinations of positive and negative values. Give students about 10 minutes to practice a few of these. Small groups would be beneficial is allowing them to “teach each other”. Now, give examples of a trinomial that has a GCF and when removed the remaining trinomial has a = 1. Then factor the trinomial. Provide students with several examples. Once again allow a few minutes to practice in their small groups. Follow up with these 2 tips: there cannot exist a common factor within the binomial if there is not a common factor in the trinomial the signs on the terms of the trinomial can assist in placement of the signs in the binomial factors 5-10 min Instructor provides a brief lecturette to reinforce the key points of the lesson and foreshadow the next lesson. Assessment Strategies/Comments: Hawkes Learning software certification. [Back to Table of Contents] Math 101: Beginning Algebra Course Kit pg 70 Developmental Studies Lesson Plan Review Session – Factoring Trinomials Initial Lesson by: Debbie Moran Course/Unit Focus: Beginning Algebra (MAT 101) /Factoring Primary Course Outcome(s): 5. Factor polynomials Learning Objectives: By the end of the session, learners will: 4.16 Factor trinomials with a coefficient of 1 4.17 Factor trinomials with a coefficient of 1 after factoring out the GCF. Duration: 50 min. Time 15 min 15 min Description/Activity Introduce Factoring an equation of the form 𝒙𝟐 + 𝒃𝒙 + 𝒄 by showing the product of two binomials and how the numerical portions are related. (𝒙 + 𝟐)(𝒙 + 𝟓) = 𝒙𝟐 + 𝟕𝒙 + 𝟏𝟎 where 7 = 2+5 and 10 = (2)(5) Now, show the reverse of this problem looking for factors of 10 that add to be 7. Include other examples with values for b and c being different combinations of positive and negative values. Give students about 10 minutes to practice a few of these. Small groups would be beneficial is allowing them to “teach each other”. Now, give examples of a trinomial that has a GCF and when removed the remaining trinomial has a = 1. Then factor the trinomial. Provide students with several examples. Once again allow a few minutes to practice in their small groups. Follow up with these 2 tips: there cannot exist a common factor within the binomial if there is not a common factor in the trinomial the signs on the terms of the trinomial can assist in placement of the signs in the binomial factors 5-10 min Instructor provides a brief lecturette to reinforce the key points of the lesson and foreshadow the next lesson. Assessment Strategies/Comments: Hawkes Learning software certification. [Back to Table of Contents] Math 101: Beginning Algebra Course Kit pg 71 Developmental Studies Lesson Plan Review Session – Final Jeopardy Initial Lesson by: Debbie Moran Course/Unit Focus: Beginning Algebra (MAT 101) / Any Primary Course Outcome(s): Any/all related to the test/final. Learning Objectives: By the end of the session, learners will: 1. Practice a range of problems on factoring (in preparation for the chapter test); 2. Work together in teams to quickly identify answers to representative problems from each category. Activity: Use the Jeopardy activity to review the factoring, solving equations and applications. Assessment Strategies/Comments: Give extra credit on the test (1 to 3 points) for each group. (?) Emphasize: Materials: Duration: 1 class (75 min) Time Description/Activity 5 min Intro: Homework challenges discussed and/or collected. Instructor introduces lesson describing its importance, relationship to algebra and key concepts and skills to be addressed. Encourage Students to break into pairs 5-10 min Summary - Instructor provides a brief lecturette to reinforce the key points of the lesson and foreshadow the next lesson and assigns homework (as needed) Assessment Strategies/Comments: Homework may be assigned in Tutorial software and/or [Back to Table of Contents] Math 101: Beginning Algebra Course Kit pg 72 Appendix A: Syllabus - Math 101 Beginning Algebra Arts and Sciences Division Developmental Studies MAT 101 Beginning Algebra Syllabus ◊ Fall 2011 Course Description: This course includes the following topics: operations with signed numbers; addition, subtraction, multiplication, and division with algebraic expressions; factoring; techniques for solving linear and fractional equations; and an introduction to graphing. In addition this course covers systems of linear equations, exponents and operations with polynomials and serves as an introduction to Algebra and its applications. Purpose and Prerequisites: MAT 101 provides a basic foundation in Algebra and serves as preparation for MAT 102. Students may also elect to take MAT 101 as a refresher course. Registration for MAT 101 requires passing MAT 032 with a grade of C (70%) or higher or by satisfactory placement test (Compass/Asset). Semester Credit Hours: 3.0 (non-credit) Learning Outcomes: The student satisfactorily completing MAT 101 will have demonstrated with a minimum of 70% accuracy on all coursework the ability to: 1. 2. 3. 4. 5. 6. 7. Perform operations with real numbers and algebraic expressions; Solve linear equations, inequalities, and formulas for specified variables; Graph linear equations and determine the equation of a line; Solve systems of two linear equations and interpret solutions; Solve application problems involving the procedures and techniques above; Simplify, evaluate, and perform operations with polynomials; Factor polynomials. Required Learning Resources (Textbooks/Materials/Online): Text: Wright, D. Franklin. Developmental Mathematics. Charleston, SC: Hawkes Learning Systems, 2011. Software: Hawkes Learning Systems Other: TI-30XS (Multiview) scientific calculator Blackboard & GTC gmail: Blackboard will be used to access online documents (syllabus, course outline), resources and announce assignments and tests in coordination with the Hawkes Learning System. GTC gmail will be used to communicate important course information. Students should regularly access both Blackboard and GTC gmail to keep up to date with course announcements and assignments. Math 101: Beginning Algebra Course Kit pg 73 Course Requirements and Evaluation: Course Outline/Schedule: A course outline/schedule will be provided that identifies specific topics covered and assignments/assessments (test/quizzes) due dates across the semester. Grading Scheme: This course will have the following types of assignment/assessments and final grades will be weighted as listed. Homework (Hawkes certification) 20% Unit tests (4 @ 11.25%) 45% Final Exam 30% Activities 5% Grades: A: 90-100% B: 80-89% C: 70-79% D: 60-69% F: 0-59% There will be no extra credit given in this class and no curving of grades. Your final grade will be the grade that you earned in the class and should reflect your knowledge of the material. If this course is required as a prerequisite to another math course, you must receive a final grade of C or higher to proceed to the next course. Homework & Certification: Homework will be done online using the Hawkes Learning System software. In this software you will study, practice and “certify” on each topic covered in this class. To complete each assignment you must demonstrate your mastery by passing the certification quiz at 80%. This will earn you 100% on your homework assignment. Each homework assignment for a class will be due at the beginning of the next class period. If you submit assignments late, you may still receive partial credit for the first 5 days after each due date, but there will be a late penalty of 10% per day subtracted from your score. After the late period is over, you will receive a zero for a missed homework assignment. Because of the time needed to install and learn to use the Hawkes software, your first week’s assignments will be due at the beginning of the first class period of the second week. Your 3 lowest homework assignment grades will be dropped. Unit Tests: You cannot make up a missed test, but one missed test score will be replaced by the final exam score. Other missed tests will receive a grade of zero. If you take all unit tests, your lowest score will be replaced by the final exam score (if it is higher). Final Exam: You must take the final exam to pass this course. The final exam will be comprehensive and cannot be exempted. Activities: Consistent participation in class activities, problem-solving practice, group work, in-class quizzes, online discussions (in Blackboard), and application experiences is a central part of your learning experience. Partial Credit Rubric (A Learning Guide & Grading Tool) Where partial credit is available on a test or exam, the following rubric will be used to award points for solutions. 0 (0%) Non-responsive Solution contains no correct information. 1 (25%) Preliminary Solution contains some correct information/elem ents, but problem is unsolved/ unstarted. 2 (50%) Beginning Solution contains evidence of understanding the key concept, and a solution has been attempted/started. 3 (75%) Developing Solution clearly exhibits use of key concept(s) to structure answer, and a solution has been proposed, there is limited nonconceptual/careless calculation errors. 4 (100%) Exemplary Solution effectively uses key concept(s) and appropriate steps/methods to structure answer and provides a correct solution with no errors. Math 101: Beginning Algebra Course Kit pg 74 How to Succeed in this Class: A Checklist Read your emails and check in on Blackboard regularly. o o Your instructor will send out communications to the class via your GTC gmail account. Announcements, assignments and grades will be posted to Blackboard periodically. Attend every class. o If you miss a class, YOU are responsible for learning the material you missed. Read the book, complete the assignments and learn the material before coming back for the next class. o Note: If you miss more than 3 classes AND have less than a 70 average, you may be administratively withdrawn from the class (note: administrative withdrawals occur about a week before the last day to withdraw). Be on time and don’t leave early. o If you must leave early, inform your instructor before class. o You must be in class for at least half the class to be counted present for the day. Bring your book and calculator to class. Participate in class activities, problem-solving and discussions. Stay focused on the class from beginning to end o Do not pack up early. o Turn your cell phone and ipod off and put any unneeded distractions away. It is important for you to be focused on math (and nothing else) while in the classroom – It will help you learn. Read your textbook and study the problems and examples it provides. Do Your Homework. o As with all college classes, plan on doing at least 2 hours of study outside class for every hour spent in class. o Use Hawkes Learning System Help features and practice opportunities to master each topic. o Homework is a required part of your learning and you need your homework scores to pass this class – so make this a regular part of your study plan. Ask for help. o As soon as you have problems - Don’t wait until it is too late to recover from these problems as you might miss your chance for doing well in (or passing) the course. o See your instructor before/after class or come by his/her office. o Go to the Aspire Learning Zone (104-357, Barton Campus) o Go to the Math Center (see locations at each campus and schedules at: http://gvltec.edu/instructional_support/). o Get a private tutor - There is free tutoring at GTC both through the ALZ on the Barton campus and Instructional Support Program (ISP). o Contact Your Academic Coach – An Academic Coach will be associated with this course. The Academic Coach is available to assist students with learning success strategies such as: study skills, time management, and accessing campus resources. Students may connect with their Academic Coach through the ASPIRE Learning Zone (104-357). The ALZ serves as the learning and support center for all Developmental Studies Students. Additional information on these programs with schedules and locations can be found at the GTC website (www.gvltec.edu) under Academic and Instructional Support/Tutoring Programs at: http://gvltec.edu/tutoring/. Math 101: Beginning Algebra Course Kit pg 75 Greenville Tech Policies and Learning Resources Greenville Tech has policies and learning resources that have been developed and designed to help learners succeed. The following documents include information and guidelines for how to access resources and complete information on time. It is important that you read through them, understand your opportunities and your responsibilities and make the most of the supportive learning environment that has been designed with your success in mind. Developmental Studies Department Policies (linked through Blackboard Course Content) Arts & Sciences Division Policies (linked through Blackboard Course Content) Important Dates Fall 2011: August 15 August 15-19 September 5 October 10-11 October 26 November 23-25 December 5 December 6-12 Classes begin Add/Drop Week Labor Day Holiday (M) No Classes Fall Break (M-T) No Classes Last Day to Withdraw from 15 Week Classes (W) Thanksgiving Holiday (W-F) No Classes Last Day of Classes (M) Final Exams (T-M) Math 101: Beginning Algebra Course Kit pg 76 Appendix C: Activities Developmental Studies Activity Title: Intercepts Submitted by: Habib Aghdami Duration: 1 class period Type: Individually or in groups Subject/Unit Focus: This activity is designed for Beginning Algebra (Math 101); the section on “Graphing Linear Equations in Two Variables” Purpose of Activity: This session is related to the concept of using intercepts to graph a linear equations in two variables, as well as vertical and horizontal lines. Objective(s) addressed: By the end of the session, learners will: 1. Find the x- and y-intercepts of a linear function. 2. Graph a linear function using the x- and y-intercepts. 3. Graph vertical and horizontal lines. Description of Activity: This activity builds skills necessary for identifying equations of diagonal, horizontal and vertical lines, also graphing linear equations by choosing the x- and y-intercepts. Materials Needed: Intercepts Activity/Discussion Sheet References: This activity is designed for Beginning Algebra (Math 101); the section on “Graphing Linear Equations in Two Variables” Support Materials: Overhead or a Computer, Answers to Intercepts Activity/Discussion Sheet Math 101: Beginning Algebra Course Kit pg 77 Intercept Activity/Discussion Sheet Name______________________________ Practice Problems 1a - 1b: Graph each linear function by finding x- and y-intercepts. 1a. 2x - 3y = -6 1b. x = 3y Practice Problems 2a - 2b: Graph each linear equation. 2a. x = 4 2b. y + 5 = 0 Math 101: Beginning Algebra Course Kit Answers Intercept Activity/Discussion Sheet Answer/Discussion to 1a 2x - 3y = -6 Step 1: Find the x- and y- intercepts. Let's first find the x-intercept. What value are we going to use for y? You are correct if you said y = 0. *Find x-int. by replacing y with 0 *Inverse of mult. by 2 is div. by 2 The x-intercept is (-3, 0). Next we will find the y- intercept. What value are we going to plug in for x? If you said x = 0, you are right. *Find y-int. by replacing x with 0 *Inverse of mult. by -3 is div. by -3 The y-intercept is (0, 2) pg 78 Math 101: Beginning Algebra Course Kit pg 79 Step 2: Find at least one more point. We can plug in any x value we want as long as we get the right corresponding y value and the function exists there. Let's put in an easy number x = 1: *Replace x with 1 *Inverse of add 2 is sub. 2 *Inverse of mult. by -3 is div. by -3 So the ordered pair (1, 8/3) is another solution to our function. Note that we could have plugged in any value for x: 5, 10, -25, ..., but it is best to keep it as simple as possible. The solutions that we found are: x y (x, y) -3 0 (-3, 0) 0 2 (0, 2) 1 8/3 (1, 8/3) Math 101: Beginning Algebra Course Kit Step 3: Plot the intercepts and point(s) found in steps 1 and 2. Step 4: Draw the graph. Answer/Discussion to 1b pg 80 Math 101: Beginning Algebra Course Kit x = 3y Step 1: Find the x- and y- intercepts. Let's first find the x-intercept. What value are we going to use for y? You are correct if you said y = 0. *Find x-int. by replacing y with 0 The x-intercept is (0, 0). Next we will find the y- intercept. What value are we going to plug in for x? If you said, x = 0 you are right. *Find y-int. by replacing x with 0 The y-intercept is (0, 0) Step 2: Find at least one more point. Since we really have found only one point this time, we better find two additional solutions so we have a total of three points. We can plug in any x value we want as long as we get the right corresponding y value and the function exists there. Let's put in an easy number x = 1: pg 81 Math 101: Beginning Algebra Course Kit pg 82 *Replace x with 1 *Inverse of mult. by 3 is div. by 3 So the ordered pair (1, 1/3) is another solution to our function. Let's put in another easy number x = -1: So the ordered pair (-1, -1/3) is another solution to our function. The solutions that we found are: x y (x, y) 0 0 (-3, 0) 1 1/3 (1, 1/3) -1 -1/3 (-1, -1/3) Step 3: Plot the intercepts and point(s) found in steps 1 and 2. Math 101: Beginning Algebra Course Kit Step 4: Draw the graph. Answer/Discussion to 2a x=4 pg 83 Math 101: Beginning Algebra Course Kit pg 84 This is in the form x = c. So, what type of line are we going to end up with? Vertical. Step 1: Find the x- and y- intercepts. AND Step 2: Find at least one more point. Since this is a special type of line, I thought I would talk about steps 1 and 2 together. It does not matter what y is, as long as x is 4. Note that the x-intercept is at (4, 0). Do we have a y-intercept? The answer is no. Since x can never equal 0, then there will be no y-intercept for this equation. Some points that would be solutions are (4, 0), (4, 1), and (4, 2). Again, I could have picked an infinite number of solutions. The solutions that we found are: x y (x, y) 4 0 (4, 0) 4 1 (4, 1) 4 2 (4, 2 ) Step 3: Plot the intercepts and point(s) found in steps 1 and 2. Math 101: Beginning Algebra Course Kit Step 4: Draw the graph. Answer/Discussion to 2b y+5=0 If you subtract 5 from both sides, you will have y = -5. It looks like it fits the form y = c. With that in mind, what kind of line are we going to end up with? Horizontal. pg 85 Math 101: Beginning Algebra Course Kit pg 86 Step 1: Find the x- and y- intercepts. AND Step 2: Find at least one more point. Since this is a special type of line, I thought I would talk about steps 1 and 2 together. It doesn't matter what x is, y is always -5. So for our solutions we just need three ordered pairs such that y = -5. Note that the y-intercept (where x = 0) is at (0, -5). Do we have a x-intercept? The answer is no. Since y has to be -5, then it can never equal 0, which is the criteria of an x-intercept. So some points that we can use are (0, -5), (1, -5) and (2, -5). These are all ordered pairs that fit the criteria of y having to be -5. Of course, we could have used other solutions, there are an infinite number of them. The solutions that we found are: x y (x, y) 0 -5 (0, -5) 1 -5 (1, -5) -1 -5 (1, -5) Step 3: Plot the intercepts and point(s) found in steps 1 and 2. Math 101: Beginning Algebra Course Kit Step 4: Draw the graph. pg 87